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PHYSICAL REVIEW E 84, 051919 (2011)

Stochastic oscillations in models of epidemics on a network of cities

G. Rozhnova,1A. Nunes,1and A. J. McKane1,2

1Centro de F´ ısica da Mat´ eria Condensada and Departamento de F´ ısica, Faculdade de Ciˆ encias da Universidade de Lisboa,

PT-1649-003 Lisboa Codex, Portugal

2Theoretical Physics Division, School of Physics and Astronomy, University of Manchester, Manchester M13 9PL, United Kingdom

(Received 12 September 2011; published 28 November 2011)

We carry out an analytic investigation of stochastic oscillations in a susceptible-infected-recovered model of

disease spread on a network of n cities. In the model a fraction fjkof individuals from city k commute to city

j, where they may infect, or be infected by, others. Starting from a continuous-time Markov description of the

model the deterministic equations, which are valid in the limit when the population of each city is infinite, are

recovered. The stochastic fluctuations about the fixed point of these equations are derived by use of the van

Kampen system-size expansion. The fixed point structure of the deterministic equations is remarkably simple:

A unique nontrivial fixed point always exists and has the feature that the fraction of susceptible, infected, and

recovered individuals is the same for each city irrespective of its size. We find that the stochastic fluctuations have

an analogously simpledynamics: Alloscillations have asingle frequency, equal to thatfound intheone-city case.

We interpret this phenomenon in terms of the properties of the spectrum of the matrix of the linear approximation

of the deterministic equations at the fixed point.

DOI: 10.1103/PhysRevE.84.051919PACS number(s): 87.10.Mn, 05.40.−a, 02.50.Ey

I. INTRODUCTION

Two of the ideas that are currently dominating the discus-

sion of modeling epidemic spread are those of stochasticity

and network structure [1–6]. Deterministic models of the

susceptible-infected-recovered (SIR) type have a long history

[7,8] and have been thoroughly investigated [9,10] along with

many extensions of the models, such as age classes or higher-

order nonlinear interaction terms. Although stochasticity, due

to random processes at the level of individuals, and networks,

eitherbetweenindividualsortownsandcities,wererecognized

early on as significant and important factors, the tendency

was to model them through computer simulations. This is not

surprising: It is rather straightforward to deal with stochastic

behavior in simulations, and similarly the analytic methods

available to investigate complex networks, especially adaptive

networks, are limited. There has also been a tendency toward

developing extremely detailed agent-based models to study

disease spread [11–14], which are the antithesis of the simple

analytic approach based on the original SIR deterministic

model.

In parallel with these developments, however, there have

been several efforts to extend analytic studies into the realm

of stochastic and network dynamics. The SIR model can be

formulated as an individual-based model (IBM) which can

form a starting point for both an analytical treatment, based on

the master equation (continuous-time Markov chain) [15,16],

and numerical simulations, based on the Gillespie algorithm

[17]. The analytical studies use the system-size expansion

of van Kampen to reduce the master equation to the set

of deterministic equations previously studied, together with

a set of stochastic differential equations for the deviations

from the deterministic result. As long as one is not too close

to the fade-out boundary, there is no need to go beyond

next-to-leading order in the expansion parameter, 1/√N,

where N is the number of individuals in the system. This

already gives results which are, in general, in almost perfect

agreement with the results of simulations [18,19].

This approach has been used to study the stochastic version

of the standard SIR model [19], the susceptible-exposed-

infected-recovered (SEIR)model[20],boththesemodelswith

annual forcing [20,21], staged models [22,23], and the pair

approximation in networked models [24,25], among others.

In this paper we extend the treatment to a metapopulation

model for disease spread, which consists of n cities (labeled

j = 1,...,n), each of which contains Nj individuals. A

fraction of the population of city k, fjk, commutes to city

j and this defines the strength of the link from node k to node

j in the network of cities. We show that the methods used in

the case of one city carry over to the case where the system

comprises a network of cities and that a surprisingly simple

set of results can be derived which allow us to make quite

general predictions for a class of stochastic networked models

of epidemics.

The starting point for our analysis is a specification of

how commuters move between cities in the network. As will

become clear, the model we arrive at does not depend on the

details of how and when these exchanges take place. We then

write transition rates for the usual SIR process, now taking

into account the city of origin of the infector and infected

individuals. From the resulting equation we can immediately

findthedeterministicequationscorrespondingtothestochastic

model when Nj→ ∞ for all j. Deterministic models of this

typebegantobeconsideredlongago[26]andtheexistenceand

stabilitypropertiesoftheendemicequilibriumwerestudiedfor

different formulations of the coupling between the cities and

of the disease dynamics [27–29]. Stochastic effects in these

systems have also been analyzed from the point of view of the

relation between spatial heterogeneity, disease extinction, and

the threshold for disease onset [27,30–32].

Some rather strong and general results on the uniqueness

and global stability of the fixed points of the deterministic

modelareknown[33].Weusetheseresultsandthengobeyond

this leading-order analysis to determine the linear stochastic

corrections that characterize the quasistationary state of the

finite system. As expected, the qualitative predictions of the

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G. ROZHNOVA, A. NUNES, AND A. J. MCKANEPHYSICAL REVIEW E 84, 051919 (2011)

deterministic model are shown to be incorrect; instead, large

stochastic cycles are found, although their form is much

simpler than might naively have been expected. We show that

this is, in part, a reflection of the special nature of the fixed

points of the deterministic model.

Theoutlineofthepaper isasfollows.InSec. IIwedescribe

the basic model and apply it to the case of two cities. The

generalizationtothen-citycaseingiveninSec.III.Theresults

for the form of the sustained oscillations are given in Sec. IV

and we conclude in Sec. V. Two appendixes contain technical

details which are too cumbersome to include in the main text.

II. TWO-CITY MODEL

In this section we formulate the model when there are only

twocities;thegeneraln-citycasedescribedinSec.IIIdoesnot

introduce any new points of principle and is easily explained

once the two-city case has been understood.

The SIR model consists of three classes of individuals: sus-

ceptibles, infected, and recovered. The number of individuals

in the three classes belonging to city j are denoted by Sj, Ij,

and Rj, respectively. We assume that births and deaths are

coupled at the individual level, so that when an individual dies

another (susceptible) individual is born. This means that the

number of individuals belonging to any one city, Nj, does not

change with time, and so the number of recovered individuals

is not an independent variable: Rj= Nj− Sj− Ij, where

j = 1,2 [19].

TherearefourprocessesintheSIRmodelwhichcausetran-

sitions to a new state: infection, recovery, death of an infected

individual, and death of a recovered individual. The death of

a susceptible individual does not cause a transition, since it is

immediately replaced by another, newborn, individual which

is, by definition, susceptible. Of the four listed processes, the

final three only involve one individual and so only involve one

city. The transition rates are [19] as follows.

(a) Recovery of an infective individual (and creation of a

recovered individual):

T1≡ T(S1,I1− 1,S2,I2|S1,I1,S2,I2) = γI1,

T2≡ T(S1,I1,S2,I2− 1|S1,I1,S2,I2) = γI2.

(b) Death of an infected individual (and birth of a suscepti-

ble individual):

(1)

T3≡ T(S1+ 1,I1− 1,S2,I2|S1,I1,S2,I2) = μI1,

T4≡ T(S1,I1,S2+ 1,I2− 1|S1,I1,S2,I2) = μI2.

(c) Death of a recovered individual (and birth of a suscep-

tible individual):

(2)

T5≡ T(S1+ 1,I1,S2,I2|S1,I1,S2,I2) = μ(N1− S1− I1),

T6≡ T(S1,I1,S2+ 1,I2|S1,I1,S2,I2) = μ(N2− S2− I2).

(3)

Here γ and μ are parameters which respectively character-

ize the rate of recovery and of birth/death.

The infection processes introduce the role of the com-

muters. We let f21be the fraction of the population from city

1 which commutes to city 2, leaving a fraction (1 − f21) of

the population as residents of city 1. Similarly, for commuters

FIG. 1. A fraction fjkof residents of city k commute to city j,

where j,k = 1,2.

from city 2, as illustrated in Fig. 1. We note that the number

of individuals in city j is Mj= (1 − fkj)Nj+ fjkNk, where

j ?= k. We do not specify the nature of the commute in

more detail and assume that the fjk are a property of

the corresponding pair of cities that defines the overall

average fraction of time that an individual from one city

spends in the other city. These coefficients are taken as a

coarse-grained measure of the demographic coupling between

the cities that will be applied to all individuals indepen-

dently of disease status and do not discriminate between

different types of stays with their typical frequencies and

durations.

To see the nature of the infective interactions that occur, we

firstfixourattentiononthoseinvolvingsusceptibleindividuals

from city 1. There are four types of term:

(i) infective residents in city 1 infect susceptible residents

in city 1;

(ii) infective commuters from city 2 infect susceptible

residents in city 1;

(iii) infective residents in city 2 infect susceptible com-

muters from city 1;

(iv) infective commuters from city 1 infect susceptible

commuters from city 1 in city 2.

The rates for these to occur according to the usual

prescription for the SIR model [19] are

(i) β (1 − f21)S1(1 − f21)I1/M1,

(ii) β (1 − f21)S1f12I2/M1,

(iii) β f21S1(1 − f12)I2/M2,

(iv) β f21S1f21I1/M2,

where β is the parameter which sets the overall rate of

infection. Adding these rates together we obtain the total

transition rate for infection of S1individuals as

?

where

c11=(1 − f21)2N1

M1

c12=(1 − f21)f12N2

M1

A similar analysis can be made for the transitions in-

volving susceptible individuals from city 2. Putting these

results together we obtain the transition rates for infection as

follows.

(d) Infection of a susceptible individual:

βc11S1I1

N1

+ c12S1I2

N2

?

,

+f2

+f21(1 − f12)N2

21N1

M2

,

M2

.

T7≡ T(S1− 1,I1+ 1,S2,I2|S1,I1,S2,I2)

= β

N1

?

c11S1I1

+ c12S1I2

N2

?

,

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T8≡ T(S1,I1,S2− 1,I2+ 1|S1,I1,S2,I2)

= β

N1

?

c21S2I1

+ c22S2I2

N2

?

,

(4)

where

c11=

(1 − f21)2

1 − f21+ f12q+

(1 − f21)f12q

1 − f21+ f12q+

(1 − f12)f21

f21+ (1 − f12)q+

(1 − f12)2q

f21+ (1 − f12)q+

f2

21

f21+ (1 − f12)q,

f21(1 − f12)q

f21+ (1 − f12)q,

f12(1 − f21)

1 − f21+ f12q,

f2

12q

1 − f21+ f12q,

c12=

(5)

c21=

c22=

and q = N2/N1. We assume that N1 and N2 are not too

different, so that q is neither very small nor very large.

ThemodelisdefinedbythetransitionsratesinEqs.(1)–(4).

It is interesting that the transitions due to infection depend on

the fractions fjk only through the constants cjk defined in

Eq. (5). Other ways of accounting for commuting individuals

would typically still give rise to the form given in Eq. (4), but

with the constants cjkdefined in a different way.

Since our counting of the ways that infection takes place

was exhaustive, we expect that the constants cjk are not

independent. It is straightforward to check that they obey the

following relations:

c11+ c12= 1,c21+ c22= 1,c12= c21q.

(6)

So there are only two independent parameters in addition to

the usual SIR parameters β, γ, and μ found in the single-

city case, and we choose these to be c12 and q = N2/N1.

Our results are given in terms of these two parameters. It is

easy to see that, for each q, the range of c12is the interval

[0,q/(q +1)] where the maximum is attained for f21+ f12=

1. While exploring the general behavior of the system we

consider the cjkindependently of the underlying microscopic

model as positive parameters that take values in the wider

admissible range defined by the constraints (6).

Having specified the model it may be investigated in

two ways as indicated in Sec. I. First, it can simulated

with Gillespie’s algorithm [17], or some equivalent method.

Second, it can be studied analytically by constructing the

master equation and performing van Kampen’s system-size

expansiononthisequation.Thisisthemainfocusofthispaper.

For notational convenience we denote the states of the system

by σ ≡ {S1,I1,S2,I2}, recalling that the number of recovered

individuals from each city may be written in terms of these

variables. The master equation gives the time evolution of

P(σ,t), the probability distribution for finding the system in

state σ at time t. It takes the form [15,16]

dP(σ,t)

dt

=

?

σ??=σ

8

?

a=1

[Ta(σ|σ?)P(σ?,t) − Ta(σ?|σ)P(σ,t)],

(7)

where Ta(σ|σ?), a = 1,...,8 are the transition rates from the

state σ?to the state σ given explicitly in Eqs. (1)–(4).

The full master equation (7) cannot be solved, but the van

Kampen system-size expansion when taken to next-to-leading

order usually gives results which are in excellent agreement

with simulations. We will see that this is also the case in the

extensions of the method which we are exploring in this paper.

The system-size expansion starts by making the following

ansatz [15]:

Sj= Njsj+ N1/2

j = 1,2.

limNj→∞Ij/Nj are the fraction of individuals from city

j which are respectively susceptible and infected in the

deterministic limit. The quantities xjand yjare the stochastic

deviations from these deterministic results, suitably scaled

so that they also become continuous in the limit of large

population sizes. The ansatz (8) is substituted into Eq. (7)

and powers of?Njon the left- and right-hand sides matched

equations of the model and the next-to-leading order linear

stochastic differential equations for xj and yj. We do not

describethemethodingreatdetail,sinceitisdescribedclearly

in van Kampen’s book [15] and in many papers, including

severalontheSIRmodel[6,19,22].Insteadweoutlinethemain

results of the approximation in the remainder of this section

and give some explicit intermediate formulas in Appendix A.

Thedeterministicequationswhicharefoundtofirstorderin

the system-size expansion can also be obtained by multiplying

Eq. (7) by S1, I1, S2, and I2in turn and then summing over all

states σ. This yields

j

xj,Ij= Njij+ N1/2

sj= limNj→∞Sj/Nj

j

yj,

(8)

where Here

and

ij=

up. The leading order contribution gives the deterministic

ds1

dt

ds2

dt

di1

dt

di2

dt

= −βs1(c11i1+ c12i2) + μ(1 − s1),

= −βs2(c21i1+ c22i2) + μ(1 − s2),

(9)

= βs1(c11i1+ c12i2) − (γ + μ)i1,

= βs2(c21i1+ c22i2) − (γ + μ)i2.

For the case of cities with equal population sizes, these have

beenpreviouslyfoundandanalyzedinRef.[28].Inthecontext

ofthepresentworkwearemainlyinterestedinthefixedpoints

of these equations. We do not discuss these here, instead we

wait until Sec. III, where the case of n cities is discussed when

we can give a more general treatment.

Of more interest to us in this paper are the variables xjand

yj which describe the linear fluctuations around trajectories

of the deterministic set of equations (9). For convenience we

introduce the vector of these fluctuations z = (x1,x2,y1,y2).

Our focus is on fluctuations in the stationary state, that is,

about the nontrivial fixed point of the deterministic equations

(whichweshowinthenextsectionisunique).Thefluctuations

obtained through the system-size expansion obey a linear

Fokker-Planck equation, which is equivalent to a set of

stochastic differential equations of the form [16]

dzJ

dt

=

4

?

K=1

AJKzK+ ηJ(t),J = 1,...,4,

(10)

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where ηJ(t) are Gaussian noise terms with zero mean which

satisfy ?ηJ(t)ηK(t?)? = BJKδ(t − t?). Since the fluctuations

are about the fixed point, the 4 × 4 matrices A and B

are independent of time, and completely characterize the

fluctuations. They are given explicitly in Appendix A.

The fluctuations are analyzed in detail in Sec. IV, when

they are also compared to the results of numerical simulations.

Before discussing this, we generalize the discussion of this

section to an arbitrary network of n cities.

III. ARBITRARY NETWORK STRUCTURE

InthissectionwegeneralizethecontentofSec.IItoncities

and also find the fixed points of the deterministic dynamics in

this case.

A. n-city model

We use the same notation as in Sec. II, labeling the cities

by j and k which now run from 1 to n. It is convenient to

introduce the quantity

?

so that the number of individuals in city j may be written as

?

k?=j

= (1 − fj)Nj+

fj=

k?=j

fkj,

(11)

Mj=

1 −

?

fkj

?

Nj+

?

?

fjkNk.

k?=j

fjkNk

k?=j

(12)

There are, once again, four types of term in the process of

infection (see Fig. 2) and we again fix our attention on those

involving susceptible individuals from city 1.

(i) Infective residents in city 1 infect susceptible residents

in city 1. This gives a rate of β(1 − f1)S1(1 − f1)I1/M1.

(ii) Infective commuters from city j, j = 2,...,n, infect

susceptible residents in city 1. This gives a rate, summing over

all j, of β(1 − f1)S1

(iii) Infective residents in city j, j = 2,...,n, infect sus-

ceptible commuters from city 1. This gives a rate, summing

over all j, of β?

susceptible commuters from city 1 in city j. This gives a total

rate of β?

?

j?=1f1jIj/M1.

j?=1(1 − fj)Ijfj1S1/Mj.

(iv) Infectivecommutersfromcityk (includingcity1)infect

j?=1fj1S1

?

k?=jfjkIk/Mj.

FIG. 2. Individuals commute between n cities, illustrated for a

particular network when n = 4.

Since the transition rates for recovery and birth/death are

simple extensions of those for two cities we can now write

the transition rates for the n-city model as:

(a) Recovery of an infective individual (and creation of a

recovered individual):

Tj≡ T(S1,I1,...,Sj,Ij− 1,...,Sn,In|σ) = γIj,

(b) Death of an infected individual (and birth of a suscepti-

ble individual):

(13)

Tn+j≡ T(S1,I1,...,Sj+ 1,Ij− 1,...,Sn,In|σ) = μIj,

(14)

(c) Death of a recovered individual (and birth of a suscep-

tible individual):

T2n+j≡ T(S1,I1,...,Sj+ 1,Ij...,Sn,In|σ)

= μ(Nj− Sj− Ij),

(d) Infection of a susceptible individual:

(15)

T3n+j≡ T(S1,I1,...,Sj− 1,Ij+ 1...,Sn,In|σ)

n

?

where σ ≡ {S1,I1,...,Sj,Ij...,Sn,In} and where j =

1,...,n. The coefficients cjkin Eq. (16) may be read off from

theterms(i)–(iv),buttheyaresufficientlycomplicatedtowrite

in full that we only list them in Appendix B. In that appendix

wealsoshowthatrelationsbetweenthecjk,analogoustothose

given in Eq. (6) for the two-city case hold, and are given by

?Nj

= β

k=1

cjkSjIk

Nk

,

(16)

cjj+

?

k?=j

cjk= 1;

ckj=

Nk

?

cjk;

j,k = 1,...,n.

(17)

So in the n-city model, there are n(n − 1)/2 independent

coupling parameters cjkand (n − 1) parameters for city sizes

in additional to the usual epidemiological parameters. Note

that if all city sizes are equal the second relation in Eq. (17)

reduces to ckj= cjk. This symmetry is used in the subsequent

analysis.

Following the same path as in Sec. II, having specified

the model by giving the transition rates, we move on to the

dynamics.TheprocessisMarkovianandsosatisfiesthemaster

equation (7) except now the sum on a goes from 1 to 4n. As

detailed in Appendix A, invoking the van Kampen ansatz (8)

gives the following deterministic equations to leading order:

dsj

dt

= −βsj

n

?

n

?

k=1

cjkik+ μ(1 − sj),

(18)

dij

dt

= βsj

k=1

cjkik− (γ + μ)ij,

where j = 1,...,n. At next-to-leading order the fluctua-

tions are found to satisfy the linear stochastic differential

equation (10), but with J,K = 1,...,2n. The two matrices

A and B are given explicitly in Appendix A. They are

independentoftime,sincetheyareevaluatedatthefixedpoints

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of the dynamics (18). For the restof this section we investigate

the fixed point structure of these equations.

B. The fixed points of the deterministic equations

The fixed points of the deterministic equations (18) are

denoted by asterisks. Adding the two sets of equations we

immediately see that

(γ + μ)i∗

j= μ(1 − s∗

j),j = 1,...,n.

j,andalsousingEq.(17),

(19)

Usingthisequationtoeliminatethei∗

one finds that

?

s∗

j

(β + γ + μ) − β

n

?

k=1

cjks∗

k

?

=(γ + μ),j =1,...,n.

(20)

Two fixed points can be found by inspection. First, suppose

one of the i∗

s∗

0.Sincethecoefficientscjkarenon-negative(seeAppendixB),

then i∗

zeroasinputintoEq.(18),inthesamewayaswedidoriginally

fori∗

cjk, then they will have no infected individuals present. From

Eq.(19)itfollowsthats∗

solution where no infection is present anywhere in this cluster

ofconnected cities.We assumethatallthecitiesareconnected

either directly or indirectly, so that i∗

Ofmoreinterestiswhatwecall“thesymmetricfixedpoint.”

This has s∗

that the i∗

i∗. Using Eq. (17), s∗and i∗are found to satisfy the equations

jis zero, for instance, i∗

?= 1. From Eq. (18) we see immediately that?n

k= 0 for all k as long as c?k?= 0. Using the i∗

?= 0. Then from Eq. (19)

k=1c?ki∗

k=

kwhich are

?,weseethataslongasthecitiesareconnectedbynonzero

k= 1forthesecities.Thisisthetrivial

k= 0,s∗

k= 1 for all k.

k= s∗, a constant, for all k. From Eq. (19) one sees

kare also independent of k, and we denote them by

s∗[(β + γ + μ) − βs∗] = (γ + μ),

(γ + μ)i∗= μ(1 − s∗),

which are the fixed point equations for the ordinary “one-city”

SIR model [7,8]. As is well known, these may be solved to

give for the nontrivial fixed point

i∗=μ[β − (γ + μ)]

Due to a remarkable theorem, we can assert that the

symmetric solution given by Eq. (22) is the only nontrivial

fixed point of the deterministic equations (18) [33]. This is

proved by finding a Liapunov function for the n-city SIR

model. In fact the result is more general than we require and

was proved for the SEIR model; in Appendix B we give the

explicit form of the Liapunov function for the SIR model

and a brief outline of the proof following the argument in

Ref. [33] for this simpler case. The theorem also tells us that

the nontrivial fixed point (22) is globally stable. Therefore, we

can now go on to study stochastic fluctuations about this well

characterized attractor.

(21)

s∗=γ + μ

β

,

β(γ + μ)

.

(22)

IV. SPECTRUM OF THE STOCHASTIC FLUCTUATIONS

Based on previous studies of stochastic fluctuations in the

SIR model in different contexts, we would expect that the

fixed point behavior predicted in the deterministic limit is

replaced by large stochastic oscillations [18,19]. In effect, the

noise due to the randomness of the processes in the IBM

sustains the natural tendency for cycles to occur and amplifies

them through a resonance effect. Since the oscillations are

stochastic, straightforward averaging will simply wipe out the

cyclicstructure;tounderstandthenatureofthefluctuationswe

needtoFouriertransformthemandthenpickoutthedominant

frequencies.

So we begin by taking the Fourier transform of the linear

stochastic differential equation Eq. (10) (generalized to the

case of n cities) to find

2n

?

where the ˜ f denotes the Fourier transform of the function

f. Defining the matrix −iωδJK− AJK to be ?JK(ω), the

solution to Eq. (23) is

K=1

(−iωδJK−AJK)˜ zK(ω)= ˜ ηJ(ω),J =1,...,2n,

(23)

˜ zJ(ω) =

2n

?

K=1

?−1

JK(ω)˜ ηK(ω).

(24)

The power spectrum for fluctuations carrying the index J

is defined by

PJ(ω)≡?|˜ zJ(ω)|2?=

2n

?

K=1

2n

?

L=1

?−1

JK(ω)BKL(?†)−1

LJ(ω).

(25)

Since? = −iωI − A,whereI isthe2n × 2nunitmatrix,and

since A and B are independent of ω, the structure of PJ(ω)

is that of a polynomial of degree 4n − 2 divided by another

polynomialofdegree4n.Theexplicitformofthedenominator

is |det?(ω)|2.

Oscillations with well-defined frequencies should show up

as peaks in the power spectrum. The structure of the power

spectrum described above—with the ratio of polynomials

of high order potentially giving rise to many maxima—

might lead us to suppose that the spectrum of fluctuations

would have a rather complex structure. In fact, numerical

simulations indicate that only a single peak is present for

a large range of parameter values. An example is shown in

Fig. 3, where typical values for measles [7,10,34] were chosen

0.2 0.4 0.6 0.8 1.0ν

0.00

0.04

0.08

0.12

P4

FIG. 3. (Color online) Power spectrum for the fluctuations of

infectivesfromsimulationofathree-citymodelwithequalpopulation

sizes plotted as a function of the frequency ν = ω/(2π) 1/y. The

spectrum shown corresponds to city 1; the spectra for the other

cities are very similar. Metapopulation model parameters: N1=

N2= N3= 106, c12= 0.06, and c13= c23= 0.02. Epidemiological

parameters: γ = 365/13 1/y, μ = 1/50 1/y, and β = 17(γ + μ).

051919-5

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G. ROZHNOVA, A. NUNES, AND A. J. MCKANEPHYSICAL REVIEW E 84, 051919 (2011)

fortheepidemiologicalparameters(wekeepthesevaluesfixed

throughout this section).

To understand how this comes about, we first note that the

number of peaks in the power spectrum is given by the form

of the denominator, |det?(ω)|2; the numerator essentially just

shiftsthe positionof these peaks somewhat. Therefore, we can

understand the number and nature of the peaks by studying

the eigenvalues of ?JK, which are those of the matrix AJK

shifted by −iω.

Each pair of complex conjugate eigenvalues of AJK, λc,λ∗

gives rise to a factor in |det?(ω)|2of the form (|λc|2− ω2)2+

[2Re(λc)ω]2, and each real eigenvalue of AJK, λr, yields a

factor of the form (λ2

associatedwithcomplexeigenvaluesλcofAJKwithsmallreal

parts,andtheirpositionisapproximatelygivenbyω ≈ Im(λc).

Inthetrivialcaseofonecity,n = 1,AJKhasapairofcomplex

conjugate eigenvalues λ±

and|λ±

for a pronounced peak for ω close to Im(λ±

fulfilled because μ, the death-birth rate, is small. This carries

over to the general n-city case since, as shown in Appendix

B, λ±

parametervaluesofFig.3thenumericalvaluesofthecommon

eigenvalue pair are λ±

to be located close to ν = Im(λ±

For large demographic coupling, the n-city system behaves

as a well mixed system comprising all the cities and we expect

to find in that limit a power spectrum similar to the case

n = 1, where each city contributes proportionally to its size

to the overall spectral density. In the opposite limit, the n

cities uncouple and we find for each city the power spectrum

of the one-city case. In order to understand why additional

peaks do not show up in simulations for intermediate coupling

strengths, it is useful to consider the case n = 2, for which

the eigenvalues of AJK can be determined analytically and

depend on a single coupling parameter c12and the ratio of the

population sizes q = N2/N1[see Eq. (6) and Appendix B].

An Argand diagram of the two pairs of eigenvalues, λ±

λ±

seen that as the coupling increases, λ±

c,

r+ ω2)2. Peaks in the power spectrum are

1with Re(λ±

1) = −βμ/[2(γ + μ)]

1| =√μ(β − γ − μ)(seeAppendixB).Theconditions

1) ≈ |λ±

1| are

1always belong to the set of eigenvalues of AJK. For the

1= −0.17 ± i 2.99, so we expect a peak

1)/(2π) ≈ 0.48 1/y.

1and

2, for the two-city model is shown in Fig. 4. It can be

2follow the circle C

6422Re λ

4

2

2

4

Im λ

FIG. 4. (Color online) An Argand diagram of the eigenvalues

for the two-city model with q = N2/N1= 3/2 and c12∈ [0,1]. The

large black dots are the common eigenvalue pair λ±

smaller dark gray (blue) and light gray (green) dots are the remaining

eigenvalues λ±

interval. The eigenvalues with Reλ−

they are found for c12> 0.15. The asterisks show the eigenvalues for

the parameter values used in Fig. 5.

1. The sets of

2computed on a uniform grid of values of c12in the

2< −6 are not shown in the plot;

centered at zero that goes through λ±

the imaginary axis. Real and imaginary parts become of the

same order for very small values of the coupling, and so we

expectthepowerspectrumtobealwaysdominatedbythepeak

associated with λ±

uncoupled case. This behavior carries over to the n-city case

with symmetric coupling, for which a complete analysis of

the eigenvalues of AJKcan also be given (see Appendix B).

In particular, it can be shown that, apart from the common

eigenvalue pair λ±

additional eigenvalue pair that behaves as a function of the

coupling parameter as described above for n = 2.

For the coupling parameter that corresponds to the values

of λ±

choice of population sizes, the infective fluctuations power

spectra for the two-city model obtained from simulations and

from Eq. (25) are shown in Fig. 5. We find a nearly perfect

match between the results of numerical simulations and the

analytical calculations. In agreement with the above argument

the power spectra of city 1 and city 2 are very similar to the

power spectrum of the one-city case, which in turn is very

similar to the spectrum shown in Fig. 3 for three cities with

small coupling. In all cases the functional form of the spec-

tral density is dominated by the peak associated with the

common eigenvalue pair λ±

power spectra PJ(ν), their ratio with respect to the one-city

case, rJ(ν), decreases as the coupling increases. For two

cities and q = 1, the power spectra P3and P4of city 1 and

city 2 are equal and the relative peak amplitudes r3,4(νmax)

decrease with the coupling strength c12down to 0.5. For other

values of q, as in Fig. 5, the different peak amplitudes in two

cities reflect the symmetry P3(ν;c12,q) = P4(ν;c12/q,1/q).

Depending on q, the ratio r3,4(ν) may become even smaller

than 0.5, but due to the symmetry that relates P3and P4, the

amplitude of at least one of these peaks is always comparable

to that of the uncoupled case. More precisely, it is easy

to check that 1 ? r3(ν;c12,q) + r4(ν;c12,q) ? 2, where the

second inequality is satisfied strictly for c12= 0 and the lower

bound corresponds to the large coupling limit c12= 1 and to

ν = νmax.

1, moving away from

1that characterizes the spectrum in the

1, AJKhas a single (n − 1)-fold degenerate

2marked with asterisks in Fig. 4 and for a certain

1. As for the amplitudes of the

0.2 0.4 0.6 0.8 1.0

0.00

0.04

0.08

0.12

P3

0.2 0.4 0.6 0.8 1.0

0.00

0.05

0.10

0.15

0.20

P4

FIG. 5. (Color online) Power spectra for the fluctuations of

infectives from simulation of the two-city model [(red) dots] and

analytic calculation (black solid curve) plotted as a function of the

frequency ν = ω/(2π) 1/y. The population sizes were chosen to be

N1= 106and N2= 1.5 × 106so that their ratio is 3/2. The coupling

coefficient c12= 0.1. The location of the eigenvalues for this choice

of parameters is indicated in Fig. 4 by asterisks and large dots.

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PHYSICAL REVIEW E 84, 051919 (2011)

The general case of three cities with no symmetry can

also, in principle, be treated analytically because finding the

eigenvalues of AJK reduces to finding the roots of a fourth

order polynomial. However, the problem now depends on

three independent coupling parameters and two parameters

for city sizes and closed form expressions are too lengthy

to be useful. An approximate, concise description of the

behavior of the eigenvalues of AJK can be given in terms

of only two parameters that measure coupling strength and

coupling asymmetry (see Appendix B). In this approximation,

we assume that all the cjk, j ?= k, are of order√μ and treat

μ as the small parameter of the system. Simple expressions

for the real parts and the absolute values of the additional

eigenvalue pairs λ±

derived [see Eqs. (B22) and (B24)]. These show that, in this

approximation, both eigenvalue pairs behave as described for

thesymmetriccase.Asthecouplingincreases,botheigenvalue

pairs follow the circle C centered at zero that goes through

λ±

imaginary parts become of the same order within the scope of

the approximation. Equation (B22) also shows how the asym-

metry lifts the degeneracy of the two pairs λ±

coupling increases, the two eigenvalue pairs move along the

circle C at different speeds. We have checked that Eqs. (B22)

and (B24) give a good approximation to the exact results in

the regime when the eigenvalues are complex.

The same behavior is illustrated in Fig. 6, where a plot of

the exact solutions for λ±

correspond to taking those of Fig. 3 and allowing one of the

coupling coefficients to span the whole admissible range. One

of the eigenvalues is shown only up to c12= 0.12, where its

real part becomes smaller than −6.

In Fig. 7 we show numerical results for the behavior of the

eigenvaluesofAJKinthecaseoffourcitieswithdifferentpop-

ulation sizes and a certain choice of the coupling coefficients

cjk, j,k = 1,2,3,4. We make use of the following notation

for the diagonal and off-diagonal coupling coefficients (see

Appendix B): cjj= 1 − ˆ cjj x√μ and cjk= ˆ cjk x√μ,

respectively. We then calculate the set of three nontrivial

eigenvalue pairs as the coupling strength x varies in a suitable

2, λ±

3of AJKup to terms of order μ can be

1, moving away from the imaginary axis. The real and

2, λ±

3. As the

2,3is shown for parameter values that

6422Re λ

4

2

2

4

Im λ

6422Re λ

4

2

2

4

Im λ

FIG. 6. (Color online) An Argand diagram of the eigenvalues for

a three-city model with equal population sizes, c12∈ [0,0.98] and the

other parameters as in Fig. 3. The large black dots are the common

eigenvalue pair λ±

gray (green) dots are the remaining eigenvalues λ±

λ±

interval. The eigenvalues with Reλ−

they are found for c12> 0.12.

1. The sets of smaller dark gray (blue) and light

2(left panel) and

3(right panel) computed on a uniform grid of values of c12in the

2< −6 are not shown in the plot,

6422Re λ

4

2

2

4

Im λ

FIG. 7. An Argand diagram of the eigenvalues for a four-city

model with the coupling strength x ∈ [0,0.52]. The large black dots

are the common eigenvalue pair λ±

λ±

are shown as sets of smaller gray dots. As in the previous

figures we only show eigenvalues whose real part is larger than

−6. Metapopulation model parameters: N2/N1:N3/N1:N4/N1=

2:3:4, ˆ c12= 1/√μ = 2ˆ c13= 5ˆ c14/2 = 5ˆ c23/2 = 3ˆ c24= 4ˆ c34.

1. The remaining eigenvalues

2,3,4computed on a uniform grid of values of x in the interval

interval, keeping the ˆ cjkfixed. These results suggest that the

behavior of the eigenvalues of AJK is essentially given by

the description of the symmetric case and that more general

couplings break the degeneracy as in the case n = 3, with no

effects in the contributions to the peaks in the power spectrum.

V. DISCUSSION AND CONCLUSIONS

In this paper we have extended the analysis of a metapop-

ulation model of epidemics into the stochastic domain.

Frequently, epidemic models involving a spatial component,

such as the interaction between several cities, are studied

purely deterministically [28,29] or through computer sim-

ulations [6,13,27]. We have demonstrated how a stochastic

metapopulation model can be studied analytically by using a

relativelystraightforwardextensionofthemethodologywhich

was used to study a well-mixed population in a single city.

We adopted a simple specification of residents and com-

muters in order to set up the model. However, the coefficients

which appear in the dynamical equations are generic and

would appear in the same form if residents and commuters

were included in a different way. It is evident that there are

many ways of characterizing the interchange of individuals

between cities which will result in the same model; only the

identification of the coefficients with the underlying structure

will be different.

Thedeterministicformofthemodelpredictsthatthesystem

willreachastablefixedpointwheretheproportionofinfected,

susceptible,andrecoveredindividualsisthesameineverycity.

Thestochasticversionofthemodelalsopredictsacleansimple

result: that the large sustained oscillations which replace the

deterministic predictions of constant behavior have a single

frequency which is the same for every city. Moreover, for

small, large, and intermediate coupling between the cities,

the form of the power spectrum of these fluctuations is closely

approximatedbythepowerspectrumofthesingle-citysystem.

It is remarkable that such a simple result occurs in what is a

quite complicated stochastic nonlinear metapopulation model.

We hope to explore the range of validity of this result and

its robustness to the addition of new features to the model in

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G. ROZHNOVA, A. NUNES, AND A. J. MCKANEPHYSICAL REVIEW E 84, 051919 (2011)

the future. In any case, we believe that the work presented

here gives a firm foundation to possible future work, including

comparisons with the data available on childhood diseases.

ACKNOWLEDGMENTS

Financial

for Science and Technology (FCT) under Contract No.

POCTI/ISFL/2/261isgratefullyacknowledged.G.R.wasalso

supported by FCT under Grant No. SFRH/BPD/69137/2010.

supportfrom the PortugueseFoundation

APPENDIX A: SYSTEM-SIZE EXPANSION

Here we give some of the key steps in the application of

the system-size expansion to the model explored in this paper.

The method has been extensively discussed in the literature

[6,15,19–25], and so we confine ourselves to a brief outline

andtodisplayingthemostimportantintermediateresultsinthe

derivation. We assume that we are carrying out the calculation

for the n-city case discussed in Sec. III; the corresponding

results for Sec. II can be obtained simply by setting n = 2.

The first point to mention is that there are apparently n

expansion parameters: {N1,...,Nn}. The method is valid if

they are all large and of the same order. More formally we

can take, for instance, N1≡ N as the expansion parameter

and express all the other Njin terms of it: Nj= Nqj, where

the qj= Nj/N, j = 2,...,n are of order one. In practice,

the method seems to work well when the qjare significantly

different from one, but this has to be checked a posteriori,

for instance, by comparing the analytic results with those

obtained using computer simulations. In what follows we do

not introduce the qjexplicitly; we simply take all the Nj’s to

be of the same order in the expansion.

The van Kampen ansatz Eq. (8) replaces the discrete

stochastic variables σ by the continuous stochastic variables

z and so we write the transformed probability distribution

P(σ,t) as ?(z,t). Since this transformation is time-dependent,

substituting the ansatz into dP/dt on the left-hand side of

Eq. (7) gives [15]

dP(σ,t)

dt

=∂?(z,t)

∂t

−

n

?

∂?(z,t)

j=1

?Nj

∂?(z,t)

∂xj

dsj

dt

−

n

?

j=1

?Nj

∂yj

dij

dt.

(A1)

The right-hand side of the master equation (7) can be put

into a form from which it is simple to apply the expansion

procedure. To do this one introduces step operators [15]

defined by

?±1

Sjf(S1,...,Sj,...,Sn,I1,...,In)

= f(S1,...,Sj± 1,...,Sn,I1,...,In),

?±1

= f(S1,...,Sn,I1,...,Ij± 1,...,In),

for a general function f and where j = 1,...,n. Using

these operators the master equation (7) may be written

(A2)

Ijf(S1,...,Sn,I1,...,Ij,...,In)

as

dP(σ,t)

dt

=

n

?

j=1

???Ij− 1?Tj+

?1

??Ij

?Sj

??Sj

− 1

?

Tn+j

?

+

?Sj

− 1

?

T2n+j+

?Ij

− 1

T3n+j

?

P(σ,t).

(A3)

Within the system-size expansion these operators have a

simple structure,

?Sj=

∞

?

p=0

N−p/2

j

p!

∂p

∂xp

j

,?Ij=

∞

?

p=0

N−p/2

j

p!

∂p

∂yp

j

,

(A4)

and so all the terms of the right-hand side of Eq. (A3) may

be straightforwardly expanded. Comparing these with the left-

hand side in Eq. (A1) the leading order (∼?Nj) yields the

order (which is of order one) gives a Fokker-Planck equation:

deterministic equations given by Eq. (18). The next-to-leading

∂?

∂t

= −

2n

?

J,K=1

∂

∂zJ

[AJKzK?] +1

2

2n

?

J,K=1

BJK

∂2?

∂zJ∂zK.

(A5)

The 2n × 2n matrices A and B which appear in this equation

have the following form. Writing A in blocks of four n × n

submatrices,

?A(1)A(2)

the elements of these submatrices are

A =

A(3)A(4)

?

,

(A6)

A(1)

jk= −μδjk− βδjk

n

?

?=1

cj?i?,

A(2)

jk= −β

?Nj

?

Nk

?1/2

sjcjk,

(A7)

A(3)

jk= βδjk

n

?=1

cj?i?,

A(4)

jk= −(μ + γ)δjk+ β

?Nj

Nk

?1/2

sjcjk.

Writing B in a similar way to A in Eq. (A6), the elements of

the submatrices are

B(1)

jk= μδjk(1 − sj) + βδjk

n

?

?=1

sjcj?i?,

B(2)

jk= B(3)

jk= −μδjkij− βδjk

n

?

?=1

sjcj?i?,

(A8)

B(4)

jk= (γ + μ)δjkij+ βδjk

n

?

?=1

sjcj?i?.

From Eqs. (A7) and (A8) it is clear that the matrices Ajkand

Bjk depend on the solutions of the deterministic equations

given in Eq. (18). However, since we are interested only

in fluctuations about the stationary state, these matrices are

evaluated at the fixed point. Since the unique stable fixed point

is the symmetric one, the same for all cities, the entries (A7)

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PHYSICAL REVIEW E 84, 051919 (2011)

and (A8) are given by

A∗(1)

jk= −[μ + βi∗]δjk,A∗(2)

jk= −β

?Nj

?Nj

Nk

?1/2

s∗cjk,

A∗(3)

jk= βi∗δjk,A∗(4)

jk= β

Nk

?1/2

s∗cjk− (μ + γ)δjk,

(A9)

and

B∗(1)

jk

B∗(2)

jk

= 2μ(1 − s∗)δjk,

= B∗(3)

B∗(4)

jk

= 2(γ + μ)i∗δjk,

(A10)

jk

= −i∗[μ + βs∗]δjk,

where we have used the fixed-point equation (21) to simplify

some of the entries in Eq. (A10).

Finally, the Fokker-Planck equation (A5) is equivalent to

the stochastic differential equation (10). We work with the

latter, since we wish to use Fourier analysis to analyze the

nature of the fluctuations, and since Eq. (10) is linear, it can

easilybeFouriertransformed,asdiscussedindetailinSec.IV.

APPENDIX B: SOME RESULTS FOR THE n-CITY CASE

In this Appendix we give some of the derivations for the

n-city case discussed in Secs. III and IV which are too long

and cumbersome to be given in the main text.

1. The coefficients cjk

The coefficients cjkappearing in Eq. (16) may be read off

from the four types of term (i)–(iv) given in Sec. III:

(1 − fj)2Nj

?(1 − fj)Nj+?

+

??=j

for j = 1,...,n and

(1 − fj)fjkNk

?(1 − fj)Nj+?

+

?

for j,k = 1,...,n and j ?= k.

To prove the first relation given in Eq. (17), consider the

sum cjj+?

Eq. (B1) combines with the last term in Eq. (B2) to give

?f?jNj+?

??

??

cjj=

m?=jfjmNm

f2

?jNj

?

?

?(1 − f?)N?+?

m?=?f?mNm

?

(B1)

cjk=

m?=jfjmNm

?

fkj(1 − fk)Nk

?(1 − fk)Nk+?

?(1 − f?)N?+?

m?=kfkmNm

f?jf?kNk

?

+

??=j,k

m?=?f?mNm

?

(B2)

k?=jcjk. The first term in Eq. (B1) combines with

the first term in Eq. (B2) to give (1 − fj). The last term in

?

??=j

f?j

?(1 − f?)N?+?

=

??=j

?

k?=j,?f?kNk

m?=?f?mNm

?

?

?

f?j

k?=?f?kNk

m?=?f?mNm

?

?(1 − f?)N?+?

fkj

?(1 − fk)Nk+?

?

?,

=

k?=j

m?=kfkmNm

m?=kfkmNm

?

(B3)

where in the last line we have performed a relabeling.

Combining the middle term of Eq. (B2) with the result in

Eq. (B3) gives

?(1 − fk)Nk+?

?

k?=j

fkj

m?=kfkmNm

m?=kfkmNm

?

?(1 − fk)Nk+?

?

= fj,

(B4)

using Eq. (11). Adding this to the result (1 − fj) found earlier

proves the result cjj+?

the interchange of j and k. Therefore,

k?=jcjk= 1.

We also note from Eq. (B2) that cjk/Nkis symmetric under

cjk

Nk

=ckj

Nj,

(B5)

which is the second relation in Eq. (17).

2. Uniqueness and stability of the fixed point

In Sec. III we asserted that the deterministic equations

(18) have a unique nontrivial fixed point, which was glob-

ally stable. Here we prove this by giving a Liapunov

function forthedynamical

ant region R = {(s1,...,sn,i1,...,in) : 0 ? sj? 1,0 ? ij?

1,sj+ ij? 1,j = 1,...,n},wherethesystemisdefined.This

isamodificationofthefunctiongiveninRef.[33]fortheSEIR

model. The proof assumes that the matrix of the coupling

coefficients cjk is irreducible, which means that any two

cities have a direct or indirect interaction. Otherwise, the

proof breaks down because the n cities may be split into

noninteracting subsets, and several equilibria may be found by

combining disease extinction in some subsets with nontrivial

equilibrium in other subsets.

Let βjk≡ βcjks∗

point of Eq. (18), and denote by M the matrix defined by

Mkj= βjk,j ?= k, and?n

equation Mv = 0 is spanned by a single vector (v1,...,vn),

vj> 0,j = 1,...,n. Let L(s1,...,sn,i1,...,in) be defined as

?

L has a global minimum in R at the fixed point. Functions of

thisformhavebeenusedintheliteratureasLiapunovfunctions

for fixed points of ecological and epidemiological models,

whose variables take only positive values [33]. Differentiating

L along the solutions of Eq. (18), and following the proof of

Theorem 1.1 in Ref. [33], we obtain

systeminthe invari-

ji∗

k, where (s∗

1,...,s∗

n,i∗

1,...,i∗

n) is a fixed

k=1Mkj= 0,j = 1,...,n. It can be

shown ( [33], Lemma 2.1) that the solution space of the linear

L =

n

j=1

vj(sj− s∗

jlnsj+ ij− i∗

jlnij).

˙L ?

n

?

j,k=1

vjMkj

?

2 −

s∗

j

sj

−sj

s∗

j

ik

i∗

k

i∗

j

ij

?

.

(B6)

The properties of the coefficients vj in the definition of L

play a crucial role in the derivation of the second term in this

inequality. Use has been made of the identity

n

?

j=1

vj

n

?

k=1

βcjks∗

jik=

n

?

j=1

vj(γ + μ)ij,

(B7)

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which, in turn, uses the fact that Mv = 0 can be written as

n

?

Following Ref. [33], it can then be shown that the right-

hand side of Eq. (B6) is strictly negative except at

(s∗

for this fixed point in R, and the fixed point is unique and

globallystable.Notethattheresultalsoholdswhenthedisease

transmissibility β, the recovery rate γ, and the birth-death rate

μ are different in different cities, in which case the nontrivial

equilibrium is, in general, not symmetric.

j=1

βckjs∗

ki∗

jvk=

n

?

j=1

βcjks∗

ji∗

kvj,k = 1,...,n.

(B8)

1,...,s∗

n,i∗

1,...,i∗

n). Therefore, L is a Liapunov function

3. Nature of the eigenvalues of the matrix A

In this section we give some results on the eigenvalues of

A which are required for the discussion in Sec. IV.

We first recall that A is closely related to the stability

matrix of the deterministic equations (18). In fact, in most

applications of the system-size expansion they are equal. In

our case because we have n expansion parameters√Nj, they

are not equal, but closely related. A simple calculation of the

Jacobian, J, from Eq. (18), shows that

S = diag(√N1,...,√Nn).

The effect of the transformation is simply that one can obtain

J from A by omitting the terms (Nj/Nk)1/2in A(2)

in Eq. (A7) or in A∗(2)

since it follows from the similarity transformation (B9) that

the eigenvalues of A are also the eigenvalues of J. So we may

study the simpler problem of finding the eigenvalues of the

Jacobian at the symmetric fixed point (22).

For orientation, let us explicitly calculate the characteristic

polynomial of the Jacobian for the cases of one city and two

cities. These are

R1(λ) = Q−1(d2λ2+ d1λ + d0),

where

J = S−1AS,

where (B9)

jkand A(4)

jk

jkand A∗(4)

jkin Eq. (A9). This is useful,

n = 1 :

(B10)

Q = γ + μ,

d0= μ(γ + μ)[β − (γ + μ)],

d2= γ + μ,d1= βμ,

(B11)

and

n = 2 :

Thus,

R2(λ) = Q−2(d2λ2+ d1λ + d0)(g2λ2+ g1λ + g0).

R2(λ) = R1(λ)Q−1(g2λ2+ g1λ + g0),

(B12)

where

g2= γ + μ,

g0= μ(γ + μ)[β − (1 − c12− c21)(γ + μ)].

WeseethatthefactorR1(λ)iscommon,whichsuggeststhatthe

pair of eigenvalues found in the one-city case might always be

present in the n-city case. This is easily proved by considering

the vector v = (v1,...,vn,vn+1,...,v2n)Twith components

satisfying vi= v and vi+n= v?for i = 1,...,n. Then the

eigenvector equation J∗v = λv reduces to that for one city

as required.

g1= βμ + (c12+ c21)(γ + μ)2,

(B13)

A similar method can be used to find the characteristic

polynomialforn ? 3citieswithequalpopulationsizes,where

the couplings are equal, that is,

cjk=

?1 − (n − 1)c,j = k,

j ?= k,c,

(B14)

where j,k = 1,...,n. We now take the components of the

vector to be v1= −v2= v, vn+1= −vn+2= v?, and vi=

vi+n= 0fori = 3,...,n.TheeigenvectorequationJ∗v = λv

now reduces to

?

?

Therefore, both solutions of

−

βμ

γ + μ+ λ

βμ

γ + μ− μ

?

?

v − (1 − nc)(γ + μ)v?= 0,

(B15)

v − [nc(γ + μ) + λ]v?= 0.

Q−1(h2λ2+ h1nλ + h0n) = 0,

(B16)

where

h2= γ + μ, h1n= βμ + nc(γ + μ)2,

h0n= μ(γ + μ)[β − (1 − nc)(γ + μ)],

are eigenvalues of J∗. This procedure can be repeated for

n − 1independentvectorswithonlyfournonzerocomponents

and the same symmetry as v. Therefore, the characteristic

polynomial of J∗, Rn(λ), factorizes as

(B17)

Rn(λ) = R1(λ)[Q−1(h2λ2+ h1nλ + h0n)]n−1.

(B18)

Finally, let us consider three cities with arbitrary coupling

and study the eigenvalues of J∗in the limit when the off-

diagonal coefficients cjkare small and of the same order. It

becomes clear that the coupling range to explore corresponds

to cjkof the order of√μ and it is convenient to introduce the

notation

?1 − ˆ cjjx√μ,

cjk(x) =

j = k,

j ?= k,

positive

ˆ cjkx√μ,

(B19)

where

Equation (B19) represents, for each choice of ˆ cjk, a family

of systems with all the off-diagonal coefficients cjk of the

same order, that reaches the zero coupling limit for x = 0.

The quantity x measures the distance to zero coupling along

each particular family, scaled by√μ. Taking into account

the properties of the matrix cjk, given by Eq. (17), the

characteristic polynomial of J∗is a polynomial of degree six

that can be expressed in terms of this distance x√μ and

of three other independent parameters. We choose these to

be ˆ cjj,j = 1,2,3. We know that this characteristic polynomial

factorizesasR1(λ)(λ4+ p3λ3+ p2λ2+ p1λ + p0),wherep3,

p2,p1,andp0aresomecoefficients.TherootsofR1(λ)arethe

pairofeigenvaluesλ±

of this family of systems. The polynomial of degree four can

j,k = 1,2,3 and

x

isa parameter.

1sharedbyallthecharacteristicequations

051919-10

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STOCHASTIC OSCILLATIONS IN MODELS OF ...

PHYSICAL REVIEW E 84, 051919 (2011)

be easily found by direct computation. For equal city sizes we

obtain for the coefficients

p3= γσ x√μ + O(μ),

p2= 2(β − γ) μ − 3/4 γ2ˆ p2x2μ + O(μ3/2),

p1= γ(β − γ)σ x μ3/2+ O(μ2),

p0= (β − γ)2μ2+ O(μ5/2),

where

(B20)

σ = ˆ c11+ ˆ c22+ ˆ c33,

ˆ p2= ˆ c2

Keepingonlytheleadingordertermsineachofthecoefficients

given by Eq. (B20) we find a simple approximate expressions

for the two additional eigenvalue pairs λ±

we find

2) = −γ

Re(λ±

(B21)

11+ ˆ c2

22+ ˆ c2

33− 2(ˆ c11ˆ c22+ ˆ c11ˆ c33+ ˆ c22ˆ c33).

2, λ±

3. In particular,

Re(λ±

4(σ + k) x√μ + O(μ),

3) = −γ

(B22)

4(σ − k) x√μ + O(μ),

where

k2= 4?ˆ c2

Assuming without loss of generality that ˆ c33? ˆ c22? ˆ c11, k2

is positive and so k is real. Note that k = 0 in the symmetric

case, and in that case (B22) coincides in the same order

of approximation with the roots of Eq. (B16) for n = 3.

The quantities σ x√μ and k x√μ that determine, in this

approximation, the real parts of the two nontrivial eigenvalue

11+ ˆ c2

22+ ˆ c2

33− ˆ c11ˆ c22− ˆ c11ˆ c33− ˆ c22ˆ c33

?.

(B23)

pairs can be interpreted as the overall coupling strength and

the coupling asymmetry for a system of family (B19). We also

find for the absolute value of the eigenvalues

?β − γ√μ + O(μ),

which shows that, for all families of the form Eq. (B19), the

eigenvaluesλ±

plane centered at zero that goes through λ±

city sizes, the same calculation can be carried out to find that

Eqs. (B22) and (B24) still hold, with Eq. (B23) replaced by

k2= σ2+1 + q21+ q31

where qjk= Nj/Nkand

˜k2= ˆ c2

|λ±

2,3| =

(B24)

2,3ofJ∗moveclosetothecircleC inthecomplex

1. For arbitrary

q21q31

˜k2,

(B25)

11+ (ˆ c22q21− ˆ c33q31)2− 2ˆ c11(ˆ c22q21+ ˆ c33q31).

(B26)

The behavior of the two nontrivial eigenvalue pairs along a

family (B19) can be described, in this approximation, in terms

of the two parameters σ and k that characterize the family

and of the scaled distance x. As x increases away from zero,

both eigenvalue pairs move along C with speeds whose ratio is

given by (σ + k)/(σ − k). The parameter k that measures the

asymmetry of the coupling causes the splittingof the two pairs

with respect to the degenerate, symmetric case. The first pair

to reach the real axis does so for x = 4√β − γ/[γ(σ + k)],

which lies within the scope of the approximation. From then

on the two real eigenvalues keep changing with x in such a

way that the square root of their product verifies the constraint

Eq. (B24) until for large x the approximation breaks down.

[1] R. Pastor-Satorras and A. Vespignani, Phys. Rev. Lett. 86, 3200

(2001).

[2] M. J. Keeling and K. T. D. Eames, J. R. Soc. Interface 2, 295

(2005).

[3] T. Gross, Carlos J. Dommar D’Lima, and B. Blasius, Phys. Rev.

Lett. 96, 208701 (2006).

[4] E. Volz and L. A. Meyers, Proc. R. Soc. B 274, 2925 (2007).

[5] V. Colizza, R. Pastor-Satorras, and A. Vespignani, Nat. Phys. 3,

276 (2007).

[6] M. Simoes, M. M. Telo da Gama, and A. Nunes, J. R. Soc.

Interface 5, 555 (2008).

[7] R.M.AndersonandR.M.May,InfectiousDiseasesofHumans:

DynamicsandControl(OxfordUniversityPress,Oxford,1991).

[8] M. J. Keeling and P. Rohani, Modelling Infectious Diseases in

Humans and Animals (Princeton University Press, Princeton,

2007).

[9] D. Schenzle, Math. Med. Biol. 1, 169 (1985).

[10] M. J. Keeling and B. T. Grenfell, Proc. R. Soc. London B 269,

335 (2002).

[11] S. Eubank, H. Guclu, V. S. A. Kumar, M. V. Marathe,

A. Srinivasan, Z. Toroczkai, and N. Want, Nature (London) 429,

180 (2004).

[12] L. A. Meyers, B. Pourbohloul, M. E. J. Newman, D. M.

Skowronski, and R. C. Brunham, J. Theor. Biol. 232, 71 (2005).

[13] C. Christensen, I. Albert, B. Grenfell, and R. Albert, Physica A

389, 2663 (2010).

[14] D. M. Aleman, T. G. Wibisono, and B. Schwartz, Interfaces 41,

301 (2011).

[15] N. G. van Kampen, Stochastic Processes in Physics and

Chemistry (Elsevier, Amsterdam, 2007).

[16] H. Risken, The Fokker-Planck Equation (Springer, Berlin,

1989).

[17] D. T. Gillespie, J. Comput. Phys. 22, 403 (1976).

[18] A. J. McKane and T. J. Newman, Phys. Rev. Lett. 94, 218102

(2005).

[19] D. Alonso, A. J. McKane, and M. Pascual, J. R. Soc. Interface

4, 575 (2007).

[20] G. Rozhnova and A. Nunes, Phys. Rev. E 82, 041906 (2010).

[21] A. J. Black and A. J. McKane, J. Theor. Biol. 267, 85 (2010).

[22] A. J. Black, A. J. McKane, A. Nunes, and A. Parisi, Phys. Rev.

E 80, 021922 (2009).

[23] A. J. Black and A. J. McKane, J. R. Soc. Interface 7, 1219

(2010).

[24] G. Rozhnova and A. Nunes, Phys. Rev. E 79, 041922 (2009).

[25] G. Rozhnova and A. Nunes, Phys. Rev. E 80, 051915 (2009).

[26] A. Lajmanovich and J. A. Yorke, Math. Biosci. 28, 221 (1976).

[27] A. L. Lloyd and R. M. May, J. Theor. Biol. 179, 1 (1996).

[28] M. J. Keeling and P. Rohani, Ecol. Lett. 5, 20 (2002).

051919-11

Page 12

G. ROZHNOVA, A. NUNES, AND A. J. MCKANEPHYSICAL REVIEW E 84, 051919 (2011)

[29] J. Arino and P. van den Driessche, Math. Popul. Stud. 10, 175

(2003).

[30] T. J. Hagenaars, C. A. Donnelly, and N. M. Ferguson, J. Theor.

Biol. 229, 349 (2004).

[31] V. Colizza and A. Vespignani, J. Theor. Biol. 251, 450

(2008).

[32] M. Barthelemy, C. Godreche, and J.-M. Luck, J. Theor. Biol.

267, 554 (2010).

[33] H. Guo, M. Y. Li, and Z. Shuai, Proc. Am. Math. Soc. 136, 2793

(2008).

[34] C. T. Bauch and D. J. D. Earn, Proc. R. Soc. London B 270,

1573 (2003).

051919-12